I had exactly the same problem with a new PC (E6300 CPU Dual Core) connected to a 5 year old LG 1915S screen. When I booted from cold (i.e. after all off for 3 or 4 hours) the BIOS logo flashed very briefly and Windows 7 flashed briefly but then the display stayed black even though I could hear the login screen tune play. Rebooting after leaving everything on for 10 minutes to get warm gave a perfectly normal startup and everything ran OK for as long as the PC was left on. The LG screen had worked fine for about 5 years on my old setup with a different motherboard and AMD 1600+ CPU.
I swapped PSU, graphics card, drives, memory chips (swapped slots) and then updated the Foxconn BIOS. Nothing had any effect. I checked and swapped cables and BIOS settings without success. I even swapped the motherboard from Foxconn to Asus as a last resort but that had no effect either.
Then (as I should have done earlier perhaps) I swapped monitors, using my daughters 2 year old 15" Dell unit on both analogue and digital connections. SUCCESS! That worked fine on both connections, so the PC boots from stone-cold with the normal BIOS and Windows screens appearing.
Conclusions? It looks as if the older LG 1915S cannot cope with changes in, or speed of, the latest motherboards and CPU. I have emailed LG and am waiting for a response but it looks to me as if I shall have to get a new screen.